Consider the shown diagram where the two chambers separated by piston–spring arrangement contain equal amounts of certain ideal gas. Initially when the temperatures of the gas in both the chambers are kept at 300 K, the compression in the spring is 1 m. The temperature of the left and the right chambers are now raised to 400 K and 500 K respectively. If the pistons are free to slide, the final compression in the spring (in cm) is 130 times x. Find value of x.

Let ${\mathcal{l}}_0$   be the initial length of each chamber,  $x_0$ be the initial compression in the spring, $P_0$  and $T_0$  be the initial pressure and temperature in each chamber. Similarly  $\mathcal{l},x,P$ and T be the respective final values. Since the chambers contain equal amounts of certain ideal gas, so
 $\frac{P_0\left(A{\mathcal{l}}_0\right)}{R{T}_0}=\frac{P_1\left(A{\mathcal{l}}_1\right)}{R{T}_1}=\frac{P_2\left(A{\mathcal{l}}_2\right)}{R{T}_2}$
  $\Rightarrow \frac{P_0{\mathcal{l}}_0}{T_0}=\frac{P_1{\mathcal{l}}_1}{T_1}=\frac{P_2{\mathcal{l}}_2}{T_2}$ .............. (1) 
                    Initially                               Finally
   
$P_{1}A=k{x}_0$  …….. (ii)
Here $P_1=P_2=P$  so
$PA=kx$  …….. (ii)
From (ii) and (iii), we get
$\frac{P}{P_0}=\frac{x}{x_0}$  ……..  (iv)
Compression in the spring is $x-x_0$  = change in the lengths of the two chambers
 $$\begin{gathered} \Rightarrow x-x_0=({\mathcal{l}}_1+{\mathcal{l}}_2)-({\mathcal{l}}_0+{\mathcal{l}}_0) \\ \Rightarrow x-x_0=\frac{P_0{\mathcal{l}}_0}{P{T}_0}(T_1+T_2)-2{\mathcal{l}}_0 \\ (\because {\mathcal{l}}_1=\frac{P_0}{P_1}\left(\frac{T_1}{T_0}\right){\mathcal{l}}_{0}and{\mathcal{l}}_2=\frac{P_0}{P}\left(\frac{T_2}{T_0}\right){\mathcal{l}}_0) \\ \Rightarrow x-x_0=\left(\frac{x_0}{x}\right)\left(\frac{{\mathcal{l}}_0}{T_0}\right)(T_1+T_2)-2{\mathcal{l}}_0 \\ (\because \frac{P_0}{P}=\frac{x_0}{x}) \\ \Rightarrow x-1=\frac{1}{x}\left(\frac{1}{300}\right)\left(900\right)-2 \\ \Rightarrow x-1=\frac{3}{x}-2 \\ \Rightarrow x=\frac{3}{x}-1 \\ \Rightarrow x^2+x-3=0 \end{gathered}$$
 Solving for x, we get X = 1.3 m = 130 cm.